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1.2 Implicit Equations

An implicit equation, such as our motivating example

figure4450

may not be expressed as a function g,

figure4456

since for x = 0, y=-1 and y=1 both satisfy our equation. All hope is not lost, as our equation may be expressed as the union of two functions:

figure4462

We may then graph each function separately, and then combine the two graphs into a single graph.

Consider the following equation:

figure4470

whose graph follows:

figure4476

If this graph were to be separated into a collection of functions, an infinite number of functions would be needed, since each function may describe at most one point for each value of x. For any value of x, an infinite number of values of y satisfy the equation given. However, for any finite region of the plane, a finite number of functions suffice. Some equations, such as

figure4564

require an infinite number of functions, even to graph finite regions of the plane, using the procedure just described.


next up previous notation contents
Next: 1.3 Relations Up: 1 Motivation Previous: 1.1 Sampling
Jeff TupperMarch 1996